Date: Mar 20, 2000 10:39 PM
Author: eppstein@euclid.ics.uci.edu
Subject: Re: unitary (Egyptian) fractions



haoyuep@aol.com (Dan Hoey) writes:
> > What is the earliest such rational that cannot be expressed as the
> > sum of n unit fractions?

> I'm sure you mean n _distinct_ unit fractions.

If it can be expressed as n unit fractions it can be expressed as n
distinct unit fractions by repeatedly using the transformation
1/(2k-1) + 1/(2k-1) ==> 1/k + 1/k(2k-1)

> By the way, I found that 732/733 has 2771 different seven-term
> representations. The largest denominator appears in the
> representation (2305193137933140 33397845 4484 45 7 3 2). The
> smallest maximum denominator appears in (26388 20524 7330 45 7 3 2).


Damn, you beat me to it. Just to solve this problem, I modified the
SmallMultiples method in my EgyptianFraction Mathematica routines
(http://www.ics.uci.edu/~eppstein/numth/egypt/Egypt.m) to use dynamic
programming instead of brute force, so now it's not hopeless to solve it
that way but it's still slow.

I posted a couple days ago the min-max denominator of any representation
(7330, with a ten-term representation) and the best denominator for a
nine-term representation (8063). I am now also able to find the best
eight-term representation:

732/733 = 1/2 + 1/3 + 1/9 + 1/20 + 1/255 + 1/8796 + 1/12461 + 1/13194
--
David Eppstein UC Irvine Dept. of Information & Computer Science
eppstein@ics.uci.edu http://www.ics.uci.edu/~eppstein/